I wonder how much software will fail on platforms where CHAR_BIT is not 8.

I worked with some MCUs whose only redeeming quality was their cost that had CHAR_BIT=16. The answer is basically everything. It made string processing was profoundly annoying, even for C.

Is it one of TI’s custom ISAs for their DSP line? The TMS430, for example, has a 16 bit char. That causes some unintuitive looking code to fight that.

No, it was with xap [1] chips. TI is also on my list of annoying chip vendors because they're one of the few companies still making big endian ARM chips, which should die in a fire.

[1] https://en.wikipedia.org/wiki/XAP_processor

I've seen 2 programs in my life that checked for CHAR_BIT (and one was a toy decimal -> binary converter). My guess is basically all of it.